MEANDERS, KNOTS, LABYRINTHS AND MAZES

January 26, 2016 2:5 WSPC/INSTRUCTION FILE6Jay25012016Jay Kappraff, Ljiljana Radović, Slavik Jablanitself. We can deform this curve without permitting it to pass through itself, i.e., nocutting. Although these deformations appear quite different, as shown in Fig.8 , theyare considered to be the same knot. If a deformation of the curve results in a simpleloop it is referred to as an unknot. To create the shadow of a knot, draw a scribbleof lines, with the restriction that at any point of intersection only two lines of thescribble intersect as shown in Fig. 9a. Notice that at each point of intersection of thescribble four edges intersect. By introducing the over/under relation in crossings ofthe shadow, we get a knot diagram. An alternating knot can be constructed fromits shadow by drawing a path through the scribble, entering a point of intersectionand taking the middle segment of the three exit choices and then proceeding alongthe path in an over-under-over-under- pattern as shown in Fig. 9b. Notice thatsome crossings, such as the crossing at point P can be eliminated by simple twistsor movements without cutting. These moves are referred to as Reidemeister movesof which there are three such unknotting rules [4]. After all such movements aremade the resulting knot can be reduced to its minimum number of crossings asshown in Fig. 9b. A minimal projection of a knot is one that minimizes the numberof crossings. This is called the crossing number, defined to be the least numberof crossings that occur in any projection of the knot. It is uniquely defined forany knot. Meander diagrams that have a minimal number of crossings are calledminimal meander diagrams.Fig. 8.A knot and its deformations.As we described in Sec. 3, because when making a closure of meander diagrams we have two possibilities, we choose the one producing a meander knotshadow without crossing lines (loops). After that by introducing under-crossingsand over-crossings along the meander knot shadow axis, we can turn it into a knotdiagram. When the crossings alternate: under-over-under-over - the knot is said tobe alternating. Given a knot, it can be transformed without cutting to eliminatecertain crossings. However unless the knot is a loop or unknot there will alwaysremain crossings, specified by the crossing number. Each knot can be classified byits crossing number.Definition 4.1. An alternating knot that has a minimal diagram in the form of aminimal meander diagram is called a meander knot.

January 26, 2016 2:5 WSPC/INSTRUCTION FILEJay25012016Meanders, Knots, Labyrinths and Mazes7Fig. 9. (a) The shadow of a knot resulting in a knot with one extraneous crossing at P; (b) theknot redrawn with the crossing at P removed.Another problem is the derivation of meander knots first introduced by S. Jablan.Several meander knots are represented here by their Gauss codes and Conwaysymbols [9,10,7]. All computations were obtained by Jablan using the programLinKnot [7]. Gauss codes of alternating meander knot diagrams can be obtained ifto the sequence 1, 2, . . . n we add a meander permutation of order n where n is anodd number and in the obtained sequence alternate the signs of successive numbers,e.g., from meander permutation (1, 8, 5, 6, 7, 4, 3, 2, 9) we obtain Gauss code{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 8, 5, 6, 7, 4, 3, 2, 9}which corresponds to rational alternating knot with nine crossings referred to by 97and also given by the Conway symbol 3 4 2. The same knot can be obtained frommeander permutations (1, 8, 7, 6, 5, 2, 3, 4, 9) and (1, 8, 7, 4, 5, 6, 3, 2, 9), giving Gausscodes{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 8, 7, 6, 5, 2, 3, 4, 9}and{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 8, 7, 4, 5, 6, 3, 2, 9}.Fig. 10.Non-isomorphic minimal meander diagrams of the knot 97 3 4 2.

January 26, 2016 2:5 WSPC/INSTRUCTION FILE8Jay25012016Jay Kappraff, Ljiljana Radović, Slavik JablanThese three representatives of the knot 3 4 2 are shown in Fig.10. It should alsobe pointed out that if an alternating knot has an alternating minimal meanderdiagram, all of its minimal diagrams need not be meander diagrams.The natural question which arises is to find all alternating meander knots withn crossings, where n is an odd number. Alternating meander knots with at mostn 9 crossings are illustrated in Fig.11Fig. 11.Alternating meander knots with at most n 9 crossings.Which knots can be represented by non-minimal meander diagrams? For example the figure-eight knot 41 , in Conway notation 2 2 with four crossings cannotbe represented by a meander diagram but can be represented by the non-minimalmeander diagram given by the Gauss code { 1, 2, 3, 4, 5, 3, 2, 1, 4, 5} withn 5 crossings. Knot 2 2 and five additional non-minimal diagrams are shown inFig. 12. You will also note that the knot is not alternating.

January 26, 2016 2:5 WSPC/INSTRUCTION FILEJay25012016Meanders, Knots, Labyrinths and Mazes9For every knot which is not a meander knot (does not have a minimal meanderdiagram) but which can be represented by some meander diagram (which is reduced,but has more crossing than the minimal diagram of that knot, i.e., more crossingthan the crossing number of that knot ), we can define its meander number, theminimum number of crossings of its meander diagrams where the minimum is takenover over all its meander diagrams. How to find knots which have meander diagram?Alternating meander knots have it, but also non-alternating knots with the sameshadows as alternating meander knots also have the meander diagram, with somecrossings change from overcrossing to undercrossing and vice versa. The next step isto make all possible crossing changes in alternating minimal meander diagrams, i.e.,in Gauss codes of alternating meander knots and see which knots will be obtained.Fig. 12.Non-minimal meander diagrams of knots 41 , 61 , 62 , 63 , 76 , and 77 .5. Two component meander linksOpen meanders with an even number of crossings offer the interesting possibility ofjoining pairs of loose ends of the meander axis, and loose ends of the meander curve.As a result, we obtain the shadow of a 2-component link with one component in theform of a circle and the other component meandering around it. A natural questionis which alternating links can be obtained from these shadows, and in general,which 2-component links have meander diagrams. It is clear that components donot self-intersect so the set of 2-component meander links coincides with the setof alternating 2-component links with non-self-intersecting components, and all oftheir minimum diagrams preserve this property.As for knots, we pose for 2-component links the natural question as to which2-component links have meander diagrams. From the definition of meander linksit is clear that the answer will be links in which both components will be knots

January 26, 2016 2:5 WSPC/INSTRUCTION FILE10Jay25012016Jay Kappraff, Ljiljana Radović, Slavik Jablanand which components are not self-crossing, i.e., it will be the shadow of a circle. In the case of alternating minimal meander diagrams, all such diagrams of2-component links will have this property. However, in the case of non-minimalmeander diagrams, some links with an odd number of crossings are represented bymeander diagrams. Moreover, their minimal diagrams have components with selfintersections, but in their non-minimal meander diagrams none of the componentshave self-intersections. Meander links up to n 10 crossings are shown in Fig. 13.Fig. 13.Meander links up to n 10 crossings.6. Sum of meander knots and linksFor two open meander sequences we can define a sum or concatenation: the operation of joining their Dyck words and connecting the second loose end of the firstwith the first loose end of the second and making a closure in order to obtain a meander knot or link diagram (see Fig. 14). The same definition extends to meanderknots and links where we concatenate the meander parts of their Gauss codes. Forparity reasons, the sum of two meander knot diagrams or the sum of two meanderlink diagrams is a meander link diagram, and the sum of a meander knot diagramand meander link diagram or vice versa is a meander knot diagram. The sum of ameander knot diagram and its mirror image is a 2-component unlink.

January 26, 2016 2:5 WSPC/INSTRUCTION FILEJay25012016Meanders, Knots, Labyrinths and Mazes11Fig. 14. Sum of a 39-crossing meander knot and 48-crossing meander link giving the 87-crossingmeander knot.7. Semi-meander or ordered Gauss code knotsIn the case of meanders the axis of a meander is infinite. If the axis is finite, we obtainsemi-meanders, where a meander curve can pass from one side of the axis to theother in a region beyond the end(s) of the axis without crossing the axis. Gauss codedepends on the choice of the initial (basic) point belonging to some arc and from theorientation of the knot. This means that every rotation or reversal of a sequence oflength 2n representing the Gauss code of a knot with n crossings represents the same(non-oriented) knot. A Gauss code will be said to be ordered if the absolute valueof the first part of its Gauss code is the sequence 1, 2, . . . n. An alternating knot willbe called an ordered Gauss code (OCG) or semi-meander knot if it has at least oneminimal diagram with an ordered Gauss code. The name semi-meander knot followsfrom the fact that the shadow of such a knot represents a meander or semi-meander.It is clear that every meander knot is OGC, and that meander knots represent theproper subset of OGC knots. For OGC knots there is no parity restriction to thenumber of crossings, so there exist OGC knots which are not meander knots, i.e.,OGC knots with an even number of crossings. Moreover, some OGC knots with anodd number of crossings are not meander knots, e.g., knot 76 2 2 1 2 which hastwo minimal diagrams, and among them only one is OGC diagram with orderedGauss code {1, 2, 3, 4, 5, 6, 7, 5, 4, 1, 2, 7, 6, 3}. Every OGC diagram iscompletely determined by the second half of its ordered Gauss code, which will becalled short Gauss code. Fig. 15 shows all semi-meander knots with n 7 crossings

January 26, 2016 2:5 WSPC/INSTRUCTION FILE12Jay25012016Jay Kappraff, Ljiljana Radović, Slavik Jablanwhich are not meander knots.Fig. 15.Semi-meander knots with n 7 crossings which are not meander knots.8. LabyrinthsAccording to the Greek myths, the skillful craftsman Daedalus created theLabyrinth. The purpose of this special architectural structure was to imprison theMinotaur, the son of Pasiphae, the wife of the Cretan King Minos. The myth of theCretan Labyrinth has been a subject of speculation and archaeological, historical,and anthropological research for a long time just as the visual representations oflabyrinthine structures concern not only art historians, but also mathematicians.Karl Kerenyi (1897-1973), the internationally renowned scholar of religion colleague of Carl Jung, and friend and advisor of Thomas Mann, returned time aftertime to the mythological research of labyrinths and interpreted them both as cultural symbols and specific geometrical structures. Right from the beginning of hislabyrinth studies, Kerenyi introduced the labyrinth from three closely interrelatedmain aspects: 1) as a mythical construction; 2) as a spiral path that was followedby dancers of a specific ritual; and 3) as a structure that was represented by a spiralline. In his 1941 essay series [15], he summarized the most important concepts ofprevious studies and made several original observations and comparisons, whichare still widely quoted and referred to in Labyrinth Studies. With the comparativemythological and morphological analysis of the Babylonian, Indonesian, Australian,Norman, Roman, Scandinavian, Finnish, English, German and medieval and Greeklabyrinth tradition, he has proven the global presence of labyrinthine structures andrevealed the artistic and architectural impulse behind the creation of them to rituals and cultic dances where participants followed aspiral line and made meanderinggestures and dance-movements. In 1963, Kerenyi devoted a lengthy essay to Greekfolk dance [16] and pointed out how the movements of the ancient labyrinth dances

January 26, 2016 2:5 WSPC/INSTRUCTION FILEJay25012016Meanders, Knots, Labyrinths and Mazes13were transformed into the main components of the Syrtos, a dance that is still performed in Greece today. And in his last book written in 1969 [17], where he exploredthe Cretan roots of the cult of Dionysis, he discussedin depth the labyrinthine andmeander-like patterns of Knossos in dance, art, and architecture. When a dancerfollows a spiral whose angular equivalent is precisely the meander, he returns to hisstarting point, wrote Kerenyi, quoting Socrates from Platos dialogue The Euthydemus. Socrates speaks there of the labyrinth and describes it as a figure whose mosteasily recognizable feature is an endlessly repeated meander or spiral line: Then itseemed like falling into a labyrinth; we thought we were at the finish, but our waybent round and we found ourselves, as it were, back at the beginning, and just as farfrom that which we were seeking at first [17].There resulted a classical picture of thisprocession, which originally led by way of concentric circles and surprising turns tothe decisive turn in the center where one was obliged to rotate on one own axis inorder to continue the circuit [17]. The labyrinths surprising turns and the decisiveturn in their center is responsible for their symbolic meaning as well. Kerenyi seesthe labyrinth as a depiction of Hades, the underworld, and interprets the structures as narrative symbols which express the existential connection between lifeand death, between the oblivion of the dead and the return of the eternal living.From a morphological perspective, Kerenyi presupposes the transformation of thespiral to the meander pattern because straight lines were easier to draw and so therounded form was early changed into the angular form. For Kerenyi the meander isthe figure of a labyrinth in linear form. In the third to second centuries BC, as heexplains, we find the figure and the word unmistakably related: in the Middle Ageslabyrinths were also called meanders [16]. We find a detailed connection betweenmeanders and labyrinths in Matthews’ book, Mazes and Labyrinths [18]. Althoughboth Matthews and Kerenyi made the connection between labyrinths and meandersclear, the ornamental evolution of angular labyrinths were not discussed by any ofthem in a way that could explain the geometrical development process underlyingthem. Our approach seeks to remedy this. Before proceeding I would like to makeclear the difference between labyrinths and mazes since these words are often usedinterchangeably. Both labyrinths and mazes can be described by graphs. However,in the case of labyrinths, there is a single path leading from the entrance to thecenter, whereas for mazes there are at various points bifurcations in the path, withsome choices of continuance leading to dead ends and others leading on to the center. So in a sense labyrinths can be thought of as being subsets of mazes in whichthere is a unicursal path through the graph.9. Labyrinth studies and visual artsWe have found that the oldest examples of geometrical ornamentation in Paleolithicart were from Mezin (Ukraine) dated to 23,000 B.C. (see Fig. 16).Among the set of ornaments found at Mezin is the first known meander friezeunder the well-known name Greek key. Take a set of parallel lines, cut a square or

January 26, 2016 2:5 WSPC/INSTRUCTION FILE14Jay25012016Jay Kappraff, Ljiljana Radović, Slavik JablanFig. 16.Fig. 17.Ornaments from Mezin.(a,b) ”Cut and paste” construction; (c) Kufic tiles.rectangular piece with the set of diagonal parallel lines incident to the first ones,rotate by 90 , and if necessary translate it in order to fit with the initial set (SeeFig. 17). More aesthetically pleasing results will be obtained by using the initial setof black and white strips of equal thickness.10. From meanders to labyrinthsThe word labyrinth is derived from the Latin word labris, making a two-sidedaxe, the motif related to the Minos palace in Knossos. The walls of the palacewere decorated by these ornaments while the interior featured actual bronze doubleaxes. This is the origin of the name labyrinth and the famous legend about Theseus,Ariadne, and the Minotaur. The Cretan labyrinth is shown on the silver coin fromKnossos (400 B.C.) as shown in Fig. 18.To create the Cretan labyrinth, first consider a Simple Alternating Transitlabyrinth, or SAT labyrinths [3,8]. An SAT labyrinth is laid out on a certain numberof concentric or parallel levels. The labyrinth is simple if the path makes essentiallya complete loop at each level, in particular, it travels on each level exactly once.It is alternating if the labyrinth -path changes direction whenever it changes level,and transit if the path progresses without bifurcation from the outside of the mazeto the center. Most SAT labyrinths occur in a spiral meander form with the path

January 26, 2016 2:5 WSPC/INSTRUCTION FILEJay25012016Meanders, Knots, Labyrinths and Mazes15leading from the outside to the center. Each such labyrinth can be sliced down itsaxis and unrolled into an open meander form. Now the path enters at the top ofthe form and exits at the bottom: the top level (center) of the labyrinth becomesthe space below the open meander form. This process is illustrated in Fig.10 forthe Cretan labyrinth. The topology of an SAT labyrinth is entirely determined byits level sequence, i.e., its open meander permutation as described in Sec. 3,forexample, the meander permutation 3,2,1,4,7,6,5. Hence the enumeration of openmeanders and their corresponding SAT labyrinths is based on the derivation ofmeander permutations. For the derivation of open meanders one can use the Mathematica program open meanders by David Bevan [13] which we modified in orderto compute open meander permutations.Fig. 18.The Cretan labyrinth.Fig. 19. Meander permutation and the unrolling procedure to create the labyrinth.How does one construct a unicursal path without knowledge of computer programs and topological transformations? The simplest natural labyrinth is a spiralmeander: a piecewise-linear equidistant spiral. It is defined by a simple algorithm:central point and after every step turn by 90 , and continue with the next step,where the sequence of step distances is 1, 1, 2, 2, 3, 3, 4, 4 . . . Tracing this sequencewe have a labyrinth path: a simple curve connecting the beginning point (the entrance) with the end point (Minotaur room) . Fig.20 shows an elegant way toconstruct a Cretan maze. Draw a black spiral meander (Fig.20a), cut out severalrectangles or squares, rotate each of them around its center by 90 , and place it backto obtain a labyrinth (Fig.20b). Even very complex labyrinths can be constructedin this way (Fig.21).It is interesting to notice that even the Knossos dancing pattern, using the shapeof a double axe, can be reconstructed in a similar way (Fig.22). So, a simple pattern(Fig 23), an optile [19], can be considered as the logo of a Paleolithic designer fromwhich Mezin ornaments can be created. These tiles were also discovered by BenNicholson who referred to them as Versatiles.

January 26, 2016 2:5 WSPC/INSTRUCTION FILE16Jay25012016Jay Kappraff, Ljiljana Radović, Slavik JablanFig. 20.Fig. 21.Cut and paste construction of a spiral labyrinth.Cut and paste construction of a complex labyrinth.Fig. 22.The Knossos dancing pattern.

January 26, 2016 2:5 WSPC/INSTRUCTION FILEJay25012016Meanders, Knots, Labyrinths and MazesFig. 23.17Prototiles.11. A Labyrinth Workshop(1) From linoleum squares of the dimensions 40 40 cm and self

Keywords: meander, labyrinth, knots, links, mazes, friezes. Mathematics Subject Classification 2000: 57M25, 01A07 1. Introduction The meander motif got its name from the river Meander, a river with many twists mentioned by Homer in the Iliad and